Literature about the "Spaltungssatz" In the german book "Lineare Algebra" by Theo de Jong, in section 6.7 the so called "Spaltungssatz" is presented (literally: splitting theorem).
It says: Let $m_\varphi = p \cdot q$ be the minimal polynomial of a linear endomorphism $\varphi : V \to V$ such that the polynomials $p, q$ are coprime (i.e. $\gcd(p,q)=1$). Then:

*

*$\ker(p(\varphi)) = \text{im}(q(\varphi))$

*$\ker(q(\varphi)) = \text{im}(p(\varphi))$

*the above two spaces are non-trivial and $\varphi$-invariant (i.e. they satisfy $\varphi(\ker(p(\varphi))) \subseteq \ker(p(\varphi))$ and analogously for the other)

*They form a direct sum $V = \ker(p(\varphi)) \oplus \text{im}(p(\varphi))= \ker(q(\varphi)) \oplus \text{im}(q(\varphi))$
Is there any other source for this theorem or an english name for it? I only have parts of the above book available, so I can't check the bibliography. And it does not occur in the other linear algebra books I checked.

Edit:
I found related StackExchange questions, concerning similar properties of the minimal polynomial.

*

*Minimal Polynomial and Invariant Subspace

*Minimal polynomial is lcm of minimal polynomials of invariant subspaces
The first mentions the additional property that $\varphi|_{\ker(p(\varphi))}$ has minimal polynomial $p$.
The second mentions the book of Hoffmann and Kunze, in which I could find a “primary decomposition theorem” which is much stronger than the Spaltungssatz of de Jong and gives more insight into the data contained in the minimal polynomial.
 A: The primary decomposition theorem in the book of Hoffmann and Kunze seems to be a fitting answer. Most of it follows by applying the Spaltungssatz of de Jong inductively. For reference I reproduce/reformulate its statement: (p. 220, Chapter 6, Theorem 12 & Corollary)
Let $\varphi : V \to V$ be a linear endomorphism. Decompose its minimal polynomial into irreducible factors (this can be done over any field). I.e. let $p_1, \dots, p_k$ be distinct irreducible monic polynomials and $n_1, \dots, n_k$ be positive integers such that $m_\varphi = \prod_{i=1}^k p_i^{n_i}$. Further, define $W_i := \ker (p_i(\varphi))$. Then the following statements hold:

*

*The spaces $W_i$ are non-trivial and $\varphi$-invariant.

*For each $i$, the restriction of $\varphi$ to $W_i$ has minimal polynomial $p_i^{n_i}$. Consequently, $\dim(W_i) \ge n_i \cdot \deg (p_i)$.

*The spaces $W_i$ form a direct sum $V = \bigoplus_{i=1}^k W_i$. Call the projections associated with this direct sum $\pi_1, \dots, \pi_k$.

*For each $i$ there is a polynomial $h_i$ such that $h_i(\varphi) =\pi_i$.

*If another linear endomorphism $\psi : V\to V$ commutes with $\varphi$, then it commutes with each $\pi_i$ and each $W_i$ is $\psi$-invariant.

*All $\varphi$-invariant subspaces can be written as direct sum of some of the $W_i$.

Parts of this can be expressed in matrix form. There exists a basis of $V$, such that the matrix of $\varphi$ wrt. this basis has block-diagonal form with non-empty blocks $A_1, \dots, A_k$ such that each $A_i$ has minimal polynomial $p_i^{n_i}$. The size of $A_i$ is at least $n_i \cdot \deg(p_i)$.
For the proof, Hoffmann and Kunze first construct the polynomials $h_i$ and derive the rest from there. The statements above which are not included in Hoffmann and Kunze are relatively easy corollaries of the others. All in all, this theorem produces a "best possible" block-diagonal form, without imposing restrictions on the underlying field. From another point of view, it characterises the relation between invariant subspaces and factors of the minimal polynomial.
